1. 1 The Open University Maths Dept University of Oxford Dept of Education Thinking Algebraically as Developing Students’ Powers John Mason OAME Toronto Feb 2009

2. 2 Ways of Working  Everything said is a conjecture –to be tested in experience –to be modified as necessary & said in order to ‘get it out’ so it can be looked at clearly and closely & said in order to ‘get it out’ so it can be looked at clearly and closely  When we disagree we offer a potential counter-example or we invite someone to modify their conjecture

3. 3 Outline  A series of tasks –Each intended to indicate a style of task and a domain of related tasks  Reflection in order to withdraw from the action –So as to promote learning from experience

4. 4 Four Consecutives Summed  Write down four consecutive numbers and add them up  and another  Now be more extreme!  What is the same, and what is different about your answers? + 1 + 2 + 3 + 6 4 Can 44 be written as the sum of 4 consecutive numbers?

5. 5 Four Consecutives Multiplies  What numbers can appear as one more than the product of four consecutive numbers? Specialisin g In order to get a sense of structure/pattern Did anyone dive in with letters?

6. 6 Sentenced: true or false? 37 + 49 – 37 = 49 Make up your own like this 3 ÷ 4 = 15 ÷ 20 Make up your own like this What is the ‘like this’ of your example? Did you calculate … ? Start calculating and then … ? See immediately?

7. 7 Doing & Undoing  What operation undoes  adding 3?  subtracting 4?  subtracting from 7?  What are the analogues for multiplication?  What operation undoes  multiplying by 3?  dividing by 2?  multiplying by 3/2?  dividing by 3/2?  dividing into 5/7?

8. 8 Sequencing Describe a construction rule for which the second and fourth pictures are as shown #2 #4 + 1 + + 2x + 1 + + 2x 1 + 2x + 2x (1 + 2x )(1 + 2x ) – 2( x2x ) 2(1+2 ) - 1 2 + 1 + 2 + 2x2 4 + 1 + 4 + 2x4 1 + 2x4 + 2x4 1 + 4x4 + 4x4 (1 + 2x2)(1 + 2x2) - 2x(2x2x2)

9. 9 Action and Awareness  Mathematical thinking involves undertaking actions  Awarenesses are what enable actions  Awarenesses trigger action depending on what is being attended to  What are the core awarenesses that lie at the heart of different mathematical topics?

10. 10 Raise Your Hand When You Can See  1/4  1/5  1/4-1/5  1/4 of 1/5  1/5 of 1/4  1/n – 1/(n+1) What do you have to do with your attention?

11. 11 Difference Divisions 4 – 2 = 4 ÷ 2 4 – 3 = 4 ÷ 3 1 2 1 2 5 – 4 = 5 ÷ 4 1 3 1 3 6 – 5 = 6 ÷ 5 1 4 1 4 7 – 6 = 7 ÷ 6 1 5 1 5 3 – 2 = 3 ÷ 2 1 1 1 1 0 – (-1) = 0 ÷ (-1) 1 -2 1 2 1 oops 1 – 0 = 1 ÷ oops 1 1 How does this fit in? Going with the grain Going across the grain

12. 12 Honsberger’s Odd Sum  See Handout!

13. 13 Powers / Imagining & Expressing / Specialising & Generalising / Conjecturing & Convincing / Ordering & Classifying / Distinguishing & Connecting / Assenting & Asserting

14. 14 Teaching Trap  Doing for the learners what they can already do for themselves  Teacher Lust: – desire that the learner learn – allowing personal excitement to drive behaviour

15. 15 Themes / Doing & Undoing / Invariance Amidst Change / Freedom & Constraint / Extending & Restricting Meaning

16. 16 For More Ideas  Developing Thinking in Algebra (Sage)  Designing & Using Mathematical Tasks (Tarquin)  Questions & Prompts for Mathematical Thinking (ATM UK) [primary and secondary versions]  Listening Counts & Listening Figures (2 vols) (Trentham)  http: //mcs.open.ac.uk/jhm3  j.h.mason @ open.ac.uk